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from Part I - Elements of differential geometry
Published online by Cambridge University Press: 30 May 2024
It is shown how the assumption of symmetry implies the Killing equations (more generally, invariance equations of arbitrary tensors are derived and discussed). It is also shown how to find the symmetry transformations of a manifold given the Killing vectors. The Lie derivative is introduced, and it is shown that the algebra of a symmetry group always has a finite dimension, not larger than n(n+1)/2, where nis the dimension of the manifold. Conformal symmetries are defined and it is shown that the algebra of the conformal symmetry group has dimension not larger than (n+1)(n+2)/2. The metric of a spherically symmetric 4-dimensional manifold is derived from the Killing equations, and its general properties are discussed. Explicit formulae for the conformal symmetries of a flat space of arbitrary dimension are given.
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